The p-widths Database


For every closed, smooth, Riemannian manifold M of dimension at least 2, there is an associated non-decreasing sequence {ωp(M)}p=1,2,... of positive real numbers. These numbers are called the p-widths of M. Here is the heuristic definition: Using geometric measure theory, we consider a certain space 𝒵 of "all objects in M of codimension 1 without boundary". By Almgren [Alm62], 𝒵 is weakly homotopy equivalent to ℝℙ∞, so its cohomology ring

H*(𝒵; ℤ2) = ℤ2[λ].

Now, a map Φ: X → 𝒵, with X being any finite simplicial complex, is called a p-sweepout if Φ*(λp) ≠ 0. Denoting by 𝒫p the set of p-sweepouts, we define the p-width ωp(M) of M as

infΦ∈𝒫p supx∈dmn(Φ) area(Φ(x)),

where dmn(Φ) denotes the domain of Φ. For a precise definition of p-widths, see Marques-Neves [MN17].


Initiated by Marques-Neves and continued by many others, there has been a spectacular development in using min-max theory to construct minimal hypersurfaces. One motivating problem is the conjecture by Yau (1982) that every Riemannian 3-manifold contains infinitely many minimal surfaces. This conjecture was solved for generic metrics by Irie-Marques-Neves [IMN18], Chodosh-Mantoulidis [CM20], X. Zhou [Zho20], and for all metrics by A. Song for [Son23]. In particular, the use of p-widths was instrumental. (As a remark, C.-Stern [CS25] provided a gluing approach to Yau's conjecture in the generic metric setting (building on Kapouleas-MaGarth [KM23]), so that the infinitely many minimal surfaces have bounded area.)


In the following, we record results regarding explicit values of p-widths. Newly added results are in red. Comments are welcome!


Spheres

Let Sn be the unit n-sphere.


Balls

Let Bn be the unit n-ball.


Projective spaces

Let RPn be the real projective space Sn/ℤ2 and CPn the complex projective space S2n+1/U(1).

ω5(RP3) > π2 

ω9(RP3) ≤ 4π [BL22]


Len spaces

Let L(p, q), with p > q, be the lens space S3/ℤp.


Bibliography